metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7⋊C32, C14.C16, C56.2C4, C28.2C8, C16.2D7, C112.3C2, C8.3Dic7, C2.(C7⋊C16), C4.2(C7⋊C8), SmallGroup(224,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C32 |
Generators and relations for C7⋊C32
G = < a,b | a7=b32=1, bab-1=a-1 >
Smallest permutation representation of C7⋊C32
►Regular action on 224 points
Generators in S224
(1 107 93 134 213 169 45)(2 46 170 214 135 94 108)(3 109 95 136 215 171 47)(4 48 172 216 137 96 110)(5 111 65 138 217 173 49)(6 50 174 218 139 66 112)(7 113 67 140 219 175 51)(8 52 176 220 141 68 114)(9 115 69 142 221 177 53)(10 54 178 222 143 70 116)(11 117 71 144 223 179 55)(12 56 180 224 145 72 118)(13 119 73 146 193 181 57)(14 58 182 194 147 74 120)(15 121 75 148 195 183 59)(16 60 184 196 149 76 122)(17 123 77 150 197 185 61)(18 62 186 198 151 78 124)(19 125 79 152 199 187 63)(20 64 188 200 153 80 126)(21 127 81 154 201 189 33)(22 34 190 202 155 82 128)(23 97 83 156 203 191 35)(24 36 192 204 157 84 98)(25 99 85 158 205 161 37)(26 38 162 206 159 86 100)(27 101 87 160 207 163 39)(28 40 164 208 129 88 102)(29 103 89 130 209 165 41)(30 42 166 210 131 90 104)(31 105 91 132 211 167 43)(32 44 168 212 133 92 106)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
G:=sub<Sym(224)| (1,107,93,134,213,169,45)(2,46,170,214,135,94,108)(3,109,95,136,215,171,47)(4,48,172,216,137,96,110)(5,111,65,138,217,173,49)(6,50,174,218,139,66,112)(7,113,67,140,219,175,51)(8,52,176,220,141,68,114)(9,115,69,142,221,177,53)(10,54,178,222,143,70,116)(11,117,71,144,223,179,55)(12,56,180,224,145,72,118)(13,119,73,146,193,181,57)(14,58,182,194,147,74,120)(15,121,75,148,195,183,59)(16,60,184,196,149,76,122)(17,123,77,150,197,185,61)(18,62,186,198,151,78,124)(19,125,79,152,199,187,63)(20,64,188,200,153,80,126)(21,127,81,154,201,189,33)(22,34,190,202,155,82,128)(23,97,83,156,203,191,35)(24,36,192,204,157,84,98)(25,99,85,158,205,161,37)(26,38,162,206,159,86,100)(27,101,87,160,207,163,39)(28,40,164,208,129,88,102)(29,103,89,130,209,165,41)(30,42,166,210,131,90,104)(31,105,91,132,211,167,43)(32,44,168,212,133,92,106), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)>;
G:=Group( (1,107,93,134,213,169,45)(2,46,170,214,135,94,108)(3,109,95,136,215,171,47)(4,48,172,216,137,96,110)(5,111,65,138,217,173,49)(6,50,174,218,139,66,112)(7,113,67,140,219,175,51)(8,52,176,220,141,68,114)(9,115,69,142,221,177,53)(10,54,178,222,143,70,116)(11,117,71,144,223,179,55)(12,56,180,224,145,72,118)(13,119,73,146,193,181,57)(14,58,182,194,147,74,120)(15,121,75,148,195,183,59)(16,60,184,196,149,76,122)(17,123,77,150,197,185,61)(18,62,186,198,151,78,124)(19,125,79,152,199,187,63)(20,64,188,200,153,80,126)(21,127,81,154,201,189,33)(22,34,190,202,155,82,128)(23,97,83,156,203,191,35)(24,36,192,204,157,84,98)(25,99,85,158,205,161,37)(26,38,162,206,159,86,100)(27,101,87,160,207,163,39)(28,40,164,208,129,88,102)(29,103,89,130,209,165,41)(30,42,166,210,131,90,104)(31,105,91,132,211,167,43)(32,44,168,212,133,92,106), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224) );
G=PermutationGroup([[(1,107,93,134,213,169,45),(2,46,170,214,135,94,108),(3,109,95,136,215,171,47),(4,48,172,216,137,96,110),(5,111,65,138,217,173,49),(6,50,174,218,139,66,112),(7,113,67,140,219,175,51),(8,52,176,220,141,68,114),(9,115,69,142,221,177,53),(10,54,178,222,143,70,116),(11,117,71,144,223,179,55),(12,56,180,224,145,72,118),(13,119,73,146,193,181,57),(14,58,182,194,147,74,120),(15,121,75,148,195,183,59),(16,60,184,196,149,76,122),(17,123,77,150,197,185,61),(18,62,186,198,151,78,124),(19,125,79,152,199,187,63),(20,64,188,200,153,80,126),(21,127,81,154,201,189,33),(22,34,190,202,155,82,128),(23,97,83,156,203,191,35),(24,36,192,204,157,84,98),(25,99,85,158,205,161,37),(26,38,162,206,159,86,100),(27,101,87,160,207,163,39),(28,40,164,208,129,88,102),(29,103,89,130,209,165,41),(30,42,166,210,131,90,104),(31,105,91,132,211,167,43),(32,44,168,212,133,92,106)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)]])
C7⋊C32 is a maximal subgroup of
D7×C32 C32⋊D7 C7⋊M6(2) C7⋊D32 D16.D7 C7⋊SD64 C7⋊Q64
C7⋊C32 is a maximal quotient of C7⋊C64
80 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 16A | ··· | 16H | 28A | ··· | 28F | 32A | ··· | 32P | 56A | ··· | 56L | 112A | ··· | 112X |
| order | 1 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | D7 | Dic7 | C7⋊C8 | C7⋊C16 | C7⋊C32 |
| kernel | C7⋊C32 | C112 | C56 | C28 | C14 | C7 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 3 | 3 | 6 | 12 | 24 |
Matrix representation of C7⋊C32 ►in GL3(𝔽449) generated by
| 1 | 0 | 0 |
| 0 | 448 | 1 |
| 0 | 396 | 52 |
| 404 | 0 | 0 |
| 0 | 5 | 211 |
| 0 | 223 | 444 |
G:=sub<GL(3,GF(449))| [1,0,0,0,448,396,0,1,52],[404,0,0,0,5,223,0,211,444] >;
C7⋊C32 in GAP, Magma, Sage, TeX
C_7\rtimes C_{32} % in TeX
G:=Group("C7:C32"); // GroupNames label
G:=SmallGroup(224,1);
// by ID
G=gap.SmallGroup(224,1);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,12,31,50,69,6917]);
// Polycyclic
G:=Group<a,b|a^7=b^32=1,b*a*b^-1=a^-1>;
// generators/relations
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